principles of lasers激光原理第2章

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Under thermal equilibrium, the blackbody continuously emits and absorb e.m. radiation power, and reaches equilibrium condition. The electromagnetic radiation called blackbody radiation. The radiation is emitted according to Planck's law, where the radiation spectrum is determined by the temperature alone, not by the body's shape or composition.
A black body in thermal equilibrium has two notable properties: 1. It is an ideal emitter: at every frequency, it emits as much energy as – or more energy than any other body at the same temperature. 2. It is a diffuse emitter: the energy is radiated isotropically, independent of direction.
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constant temperature T.
The energy density (per volume) of the e.m. radiation inside the cavity is:
=
1 2 1 E H 2 2 2
f (t )
1 T f (t )dt 0 T
applied optical fields. In the absence of a time-varying potential, (r , t ) (r )exp j t , where the wave function E
It was solved by Planck, and the blackbody theory becomes one of the
fundamental bases of modern physics.
Light mode of a cavity
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General electric field solution of Maxwell’ equestions E r , t E exp jt jkr 0
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/ (abd )
The volume of modes with frequencies lie in (0,v):
3 1 4 2 v 8 3 cn
/c
/b
/a
kz
The mode volume density per unit frequency:
independent on the cavity shape, nature of walls and medium inside the cavity. The question is what is the solution of the spectral energy density v (v, T ) ?
(r , t ) (r , t ) V (r , t )(r , t ) j 2m t
2 2
Where V(r,t) is the potential characterizing the environment of the particle, including contributions from externally


k k x nx k y n y k z n z r rx nx ry n y rz nz
A finite space supports only standing waves, thus
al
x
2
;b m
y
2
;d n
z
2

z
y
kx l

; k y m ; kz n a b d
8 v 2 hv v 3 cn exp(hv / kT ) 1
Chapter 2_L3
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Blackbody radiation theory Spontaneous emission, absorption,
and stimulated emission
Schrödinger equation Nhomakorabean
Average photon number per mode
1 n exp(hv / kT ) 1
Average energy per mode
Steady-state solution:
dN2 / dt 0
N2 n N1 1 n
hv E nhv exp(hv / kT ) 1
E (t ) E0 exp( jt ) E ( z, t ) E0 exp( jt jkz )
In a rectangular cavity, with perfectly conducting walls, filled with uniform dielectric medium. The boundary condition requires the electric field on the wall is zero, which leads to the standing wave solution. Cavity mode: a standing wave solution in a cavity with a definite wave vector k is a mode.
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N2 E E1 hv exp 2 exp N1 kT kT
Average photon number per mode
dN 2 N 2 nN 2 nN1 dt sp sp sp
Spon. Stim. Abs.
Ref: B. Saleh and M. Teich, Fundamentals of Photonics, Wiley, 2007
Einstein coefficients for the emissions
emission all occur in the cavity, when the system reaches thermodynamic equilibrium:
An approximate realization of a black surface is a hole in the wall of a large enclosure at a fixed temperature T.
Ref: Wikipedia
Blackbody radiation
Consider a cavity filled with a homogeneous and isotropic medium, kept at a
A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. (A white body is one with a "rough surface that reflects all incident rays completely and uniformly in all directions.“)
The spectral energy density v is the energy density per unit frequency range
= v dv
0

is dependent only on the temperature T, while v only on T and the frequency v,


2 k k x2 k y k z2
d a
b x
l m n a b d
2
2
2
Mode volume density
The volume of one wave vector unit in the wave vector space is 3 kz ky
What is black body? Mode density in a blackbody cavity Mode energy in a blackbody cavity Planck radiation law (spectral energy density)
Black body
W21
W12
E1
Where A is Einstein A coefficient, B21 and B12 are Einstein B coefficients.
v 0

A B12 exp(hv0 / kT ) B21
B12 B21 B W12 W21 W A 8 hv03 B cn 3
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In the framework of blackbody radiation, spontaneous emission, absorption and stimulated
AN W21N W12 N
e 2 e 2